components which are either constant or sinusoidal in nature. To
simplify the development, we shall consider a particular disturbance signal, say w*(t), and a particular reference signal, say r*(t), to be represenative signals from a given class of functions. Once the
controller is derived with respect to these signals, the results can be generalized to cover a class of functions for which r*(t) and w*(t) are assumed to belong.
We now make the following assumptions:
(A.1) For some chosen reference signal r*(t) and a particular
disturbance w*(t) there exists an open-loop control u*(t) and an
initial state x*(O) = x* such that
x *t) = f(x*(t), u*(t), w*(t))
y*(t) = Hx*(t) = r*(t) (2-3)
e(t) = r*(t) - y*(t) = 0 for all t > 0
(A.2) The elements of both x*(t) and u*(t) satisfy the scalar, linear
differential equation
(.)(r) + ar-l(.)(r-1) + . + a1(.)(1) + a = 0 (2-4)
where the characteristic roots of (2-4) are all in the closed right half-plane.
The first assumption is merely a way of stating that it is possible
to provide output tracking. A typical example where (A.1) would not hold is for a system having more outputs than inputs. This particular